Name: Pair of Tangent, Chord of Contact
Uploaded: 2019-07-19
Duration: 19 min 37 s
Description: This video contains all the details of the chord of the contact pair of tangent and cord bisected at a point. This concept is very useful for the students.

Question

Find the equations of the tangents to the ellipse

4 x^{2} + 3 y^{2} = 5

, which are parallel to the straight line

y = 3 x + 7

. Also, find the coordinates of the points of contact of the tangents which are inclined at

60 °

to the axis of

x

.

Accepted Answer

The equation of the ellipse is given as,

$4 x^{2} + 3 y^{2} = 5$

$\frac{x^{2}}{\frac{5}{4}} + \frac{y^{2}}{\frac{5}{3}} = 1 \dots (1)$

Hence, we get,

$a^{2} = \frac{5}{4}$

$b^{2} = \frac{5}{3}$

Hence, we get $b$ is greater than $a$ .

The equation of any line parallel to $y = 3 x + 7$ is given as,

$y = 3 x + c \dots (2)$

If equation $(2)$ is a tangent to equation $(1)$ , then we get,

$c = \pm \sqrt{(a^{2} m^{2} + b^{2})}$

$c = \pm \sqrt{(\frac{5}{4} \times 9 + \frac{5}{3})}$

$c = \pm \sqrt{(\frac{155}{12})}$

$c = \pm \frac{1}{2} \sqrt{\frac{155}{3}}$

Substituting the value of $c$ in equation $(2)$ , we get the required tangents on the given ellipse as,

$y = 3 x \pm \frac{1}{2} \sqrt{\frac{155}{3}}$

We know that, $y = m x \pm \sqrt{(a^{2} m^{2} + b^{2})}$ is always a tangent to the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ in the slope form.

Here, we have, $m = \tan 60 °$

$m = \sqrt{3}$

Therefore, equation of tangent to the ellipse can be given as,

$y = \sqrt{3} x \pm \sqrt{(3 \times \frac{5}{4} + \frac{5}{3})}$

$y = \sqrt{3} x \pm \sqrt{\frac{65}{12}} . . . (3)$

To solve the second part of the question, consider the $(x_{1}, y_{1})$ be the coordinates of the point of contact.

We know that, the equation of the tangent to an ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ at the point $(x_{1}, y_{1})$ is $\frac{x x_{1}}{a^{2}} + \frac{y y_{1}}{b^{2}} = 1$ .

Hence, the equation of the tangent to an ellipse $\frac{x^{2}}{\frac{5}{4}} + \frac{y^{2}}{\frac{5}{3}} = 1$ at the point $(x_{1}, y_{1})$ is $\frac{x x_{1}}{\frac{5}{4}} + \frac{y y_{1}}{\frac{5}{3}} = 1$

$\Rightarrow 4 x x_{1} + 3 y y_{1} = 5 . . . (4)$

Since, equations $(3) & (4)$ represent the same line, therefore

$\frac{4 x_{1}}{\sqrt{3}} = \frac{3 y_{1}}{- 1} = \frac{5}{\pm \sqrt{\frac{65}{12}}}$

Hence, the coordinates of the point of contact of the tangents are $(\frac{3}{2} \sqrt{\frac{5}{13}}, - 2 \sqrt{\frac{5}{39}})$ and $(\frac{- 3}{2} \sqrt{\frac{5}{13}}, 2 \sqrt{\frac{5}{39}})$ .

Find the equations of the tangents to the ellipse $4 x^{2} + 3 y^{2} = 5$ , which are parallel to the straight line $y = 3 x + 7$ . Also, find the coordinates of the points of contact of the tangents which are inclined at $60 °$ to the axis of $x$ .

Important Questions on The Ellipse

Learn and Prepare for any exam you want !

Find the equations of the tangents to the ellipse 4x2+3y2=5, which are parallel to the straight line y=3x+7. Also, find the coordinates of the points of contact of the tangents which are inclined at 60° to the axis of x.

Important Questions on The Ellipse

Learn and Prepare for any exam you want !

Find the equations of the tangents to the ellipse $4 x^{2} + 3 y^{2} = 5$ , which are parallel to the straight line $y = 3 x + 7$ . Also, find the coordinates of the points of contact of the tangents which are inclined at $60 °$ to the axis of $x$ .