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

A circle and a parabola intersect at four points; show that the algebraic sum of the ordinates of the four points is zero. Also, show that the line joining one pair of these four points and the line joining the other pair are equally inclined to the axis.

Accepted Answer

Let the equation to the circle be

$x^{2} + y^{2} + 2 g x + 2 f y + c = 0 \dots (i)$

And, let the equation of the parabola be,

$y^{2} = 4 a x . . . (i i)$

As both the curves intersect each other, on solving equations $(i) & (i i)$ , we get,

$\frac{y^{4}}{16 a^{2}} + y^{2} (1 + \frac{g}{2 a}) + 2 f y + c = 0 \dots (i i i)$

The above equation is a fourth-degree equation in $y$ . Let its roots be $y_{1}, y_{2}, y_{3}$ and $y_{4}$ , respectively. Then, we get,

$Sum of the roots = - \frac{coefficient of y^{3}}{coefficient of y^{4}}$

Hence, we get,

$y_{1} + y_{2} + y_{3} + y_{4} = 0$

Thus, sum of the ordinates of the four points at which the circle and the parabola intersect is zero.

Again, let the points of intersection be $A, B, C$ and $D$ , respectively, and let their coordinates be $(x_{1}, y_{1}), (x_{2}, y_{2}), (x_{3}, y_{3})$ and $(x_{4}, y_{4})$ , respectively.

As all these points lie on the curve $y^{2} = 4 a x$ , we get,

$y_{1}^{2} - y_{2}^{2} = 4 a (x_{1} - x_{2})$ $\Rightarrow \frac{y_{1} - y_{2}}{x_{1} - x_{2}} = \frac{4 a}{y_{1} + y_{2}} . . . (i v)$

Now, the slope of the line $A B$ , can be given as,

$m_{1} = \frac{y_{1} - y_{2}}{x_{1} - x_{2}}$

$m_{1} = \frac{4 a}{y_{1} + y_{2}} . . . (v)$

Similarly, the slope of the line $C D$ , can be given as,

$m_{2} = \frac{y_{3} - y_{4}}{x_{3} - x_{4}}$

$m_{2} = \frac{4 a}{y_{3} + y_{4}} . . . (vi)$

As the sum of $y_{1}, y_{2}, y_{3}$ and $y_{4}$ is $0$ , we get,

$y_{3} + y_{4} = - (y_{1} + y_{2})$

Substituting the value of the above equation in equation $(vi)$ , we get,

$m_{2} = \frac{4 a}{- (y_{1} + y_{2})}$

$m_{2} = - m_{1}$

So $m_{1}$ and $m_{2}$ , are equal in magnitude.

Hence, $A B$ and $C D$ , are equally inclined to the axis.

A circle and a parabola intersect at four points; show that the algebraic sum of the ordinates of the four points is zero. Also, show that the line joining one pair of these four points and the line joining the other pair are equally inclined to the axis.

Important Questions on Conic Sections. The Parabola

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