Name: Range of Quadratic Expression
Uploaded: 2019-07-19
Duration: 26 min
Description: In this interesting video, we will understand how to find the range of quadratic expression using some interesting examples. Come, let's learn more!

Question

The line $7 y - x = 25$ cuts the curve $x^{2} + y^{2} = 25$ at the points $A$ and $B$ . Find the equation of the perpendicular bisector of the line $A B$ .

Accepted Answer

Given line is $7 y - x = 25$ i.e,, $x = 7 y - 25$ and curve is $x^{2} + y^{2} - 25 = 0$

Substituting line equation in curve, we get

${(7 y - 25)}^{2} + y^{2} - 25 = 0$

$\Rightarrow 625 - 350 y + 49 y^{2} + y^{2} - 25 = 0$

$\Rightarrow 50 y^{2} - 350 y + 600 = 0$

$\Rightarrow 50 (y^{2} - 7 y + 12) = 0$

$\Rightarrow (y^{2} - 4 y) + (- 3 y + 12) = 0$

$\Rightarrow y (y - 4) - 3 (y - 4) = 0$

$\Rightarrow (y - 3) (y - 4) = 0$

$\Rightarrow y = 4, y = 3$

For $y = 4, x = 28 - 25 = 3$

For $y = 3, x = 21 - 25 = - 4$

Therefore, points $A$ and $B$ are $(3, 4) & (- 4, 3)$ .

Midpoint of line $A B = (\frac{3 - 4}{2}, \frac{4 + 3}{2})$

$= (- \frac{1}{2}, \frac{7}{2})$

Gradient of $A B = m = \frac{3 - 4}{- 4 - 3} = \frac{1}{7}$

Gradient of line perpendicular to $A B = \frac{- 1}{m} = - 7$

Equation of the perpendicular bisector of the line $A B =$ equation of line with slope $- 7$ and passing through $(- \frac{1}{2}, \frac{7}{2})$ i.e., $y - \frac{7}{2} = - 7 (x + \frac{1}{2})$

$\Rightarrow \frac{2 y - 7}{2} = \frac{- 14 x - 7}{2}$

$\Rightarrow 2 y + 14 x = 0$

$\Rightarrow y + 7 x = 0$

The line $7 y - x = 25$ cuts the curve $x^{2} + y^{2} = 25$ at the points $A$ and $B$ . Find the equation of the perpendicular bisector of the line $A B$ .

Important Questions on Quadratics

Learn and Prepare for any exam you want !

The line 7y-x=25 cuts the curve x2+y2=25 at the points A and B. Find the equation of the perpendicular bisector of the line AB.

Important Questions on Quadratics

Learn and Prepare for any exam you want !

The line $7 y - x = 25$ cuts the curve $x^{2} + y^{2} = 25$ at the points $A$ and $B$ . Find the equation of the perpendicular bisector of the line $A B$ .