If the chord of contact of P(x_1, y_1) with r | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) The equation of the chord of contact of $P(x_1, y_1)$ with respect to the circle $x^2+y^2=a^2$ is given by $xx_1 + yy_1 = a^2$. Let the circle be

Vedclass · Accepted Answer

$2a^2$

Vedclass · Answer

(B) The equation of the chord of contact of $P(x_1, y_1)$ with respect to the circle $x^2+y^2=a^2$ is given by $xx_1 + yy_1 = a^2$. Let the circle be $S: x^2+y^2-a^2=0$. The equation of the pair of tangents from $P$ to the circle is $SS_1 = T^2$,where $S_1 = x_1^2+y_1^2-a^2$ and $T = xx_1+yy_1-a^2$. This simplifies to $(x^2+y^2-a^2)(x_1^2+y_1^2-a^2) = (xx_1+yy_1-a^2)^2$. Since $\angle AOB = 90^{\circ}$,the lines $OA$ and $OB$ are perpendicular. For a circle $x^2+y^2+2gx+2fy+c=0$,the condition for perpendicular lines at the origin is $coeff(x^2) + coeff(y^2) = 0$. Expanding the equation: $(x^2+y^2-a^2)(x_1^2+y_1^2-a^2) = x^2x_1^2 + y^2y_1^2 + a^4 + 2xxy_1y - 2x^2x_1a^2 - 2yy_1a^2$. Sum of coefficients of $x^2$ and $y^2$ is $(x_1^2+y_1^2-a^2) - x_1^2 + (x_1^2+y_1^2-a^2) - y_1^2 = 0$. $x_1^2+y_1^2-a^2-x_1^2 + x_1^2+y_1^2-a^2-y_1^2 = 0$. $x_1^2+y_1^2-2a^2 = 0$,which gives $x_1^2+y_1^2 = 2a^2$.

If the chord of contact of $P(x_1, y_1)$ with respect to the circle $x^2+y^2=a^2$ meets the circle at $A$ and $B$; and if $\angle AOB=90^{\circ}$,then $x_1^2+y_1^2=$

Similar Questions

The length of the common chord of the circles $x^2+y^2-6x+5=0$ and $x^2+y^2+4y-5=0$ is:

The equation of the common chord of the circles $x^2+y^2+2x+3y+1=0$ and $x^2+y^2-5x-6y+4=0$ is

The equation of the pair of tangents drawn from the origin to the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ is:

$A$ pair of tangents is drawn from every point on the line $3x + 4y = 12$ to the circle $x^2 + y^2 = 4$. Their variable chord of contact always passes through a fixed point whose coordinates are:

Two circles which touch both the coordinate axes intersect at the points $A$ and $B$. If $A=(1,2)$,then $AB=$

Vedclass Test Series

Exam Paper Generator

Online Exam Module