Animated Solution for Mathematics - Circles: A circle with centre $(2,3)$ and radius 4 intersects the line $x+y=3$ at the points $P$ and $Q$. If the tangents at $P$ and $Q$ intersect at the point $S(\alpha ,\beta )$, then $4\alpha -7\beta$ is equal to

Imagine you are standing on the coordinate plane, looking at a circle centered at $(2,3)$ with a radius of $4$. A line, $x+y=3$, slices through this circle, creating a chord $PQ$.

Two tangents are drawn to the circle at points $P$ and $Q$, meeting at an external point $S(\alpha ,\beta )$. We aim to find the value of $4\alpha -7\beta$.

The Chord of Contact

The most powerful tool in our arsenal for this problem is the concept of the Chord of Contact. When you draw tangents to a circle from an external point $S$, the line connecting the points of tangency ($P$ and $Q$) is defined by the equation $T=0$.

For a circle given by $x^2+y^2+2gx+2fy+c=0$, the equation of the chord of contact for a point $S(x_1,y_1)$ is:

This formula acts as a shortcut, bypassing the need to calculate the specific coordinates of $P$ and $Q$.

Before applying the $T=0$ condition, we must express the circle in its general form. Starting with the standard equation:

Expanding this yields $x^2-4x+4+y^2-6y+9=16$. Simplifying, we arrive at the general form:

Substituting the point $S(\alpha ,\beta )$ into the $T=0$ formula, we obtain:

Rearranging this equation into the standard linear form $Ax+By+C=0$, we get:

This represents the line $PQ$. Since the problem defines the line $PQ$ as $x+y-3=0$, these two equations must be proportional. We set up the following ratios:

From the first two ratios, we have $\alpha -2=\beta -3$, which simplifies to:

Equating the first and third ratios, we have $\alpha -2=\frac{2\alpha +3\beta +3}{3}$. Cross-multiplying by $3$ yields:

Substituting $\alpha =\beta -1$ into $\alpha -3\beta =9$, we get $(\beta -1)-3\beta =9$, which leads to $-2\beta =10$, or $\beta =-5$. Consequently, $\alpha =-6$.

We have determined the coordinates of point $S$ to be $(-6,-5)$. The final step is to calculate the value of $4\alpha -7\beta$:

The final result is 11.